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On the Cauchy problem for a one-dimensional conservation law with initial conditions coinciding with a power or exponential function at infinity. (English. Russian original) Zbl 1501.35254

Differ. Equ. 58, No. 3, 304-313 (2022); translation from Differ. Uravn. 58, No. 3, 309-318 (2022).
The author constructs locally bounded entropy solutions to the Cauchy problem for the scalar conservation law \(u_t+|u|^{\alpha-1}u_x=u_t+(|u|^{\alpha-1}u/\alpha)_x=0\), \(\alpha>1\). The initial function is supposed to be either a power or exponential nonnegative function. The constructed solutions are piecewise smooth and contain a countable family of discontinuity lines going from \(-\infty\). Surprisingly, the constructed solutions change their sign after passing through each discontinuity. Moreover, it is shown that a nonnegative entropy solution does not actually exist.

MSC:

35L03 Initial value problems for first-order hyperbolic equations
35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
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